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Figure 18. Time evolutions of the (a) vertical velocity (ms_1), (c) potential temperature anomaly (K), and (e) moisture anomaly (kg kg-1) from the one-dimensional cloud model, and the averaged (b) vertical velocity (m s~1), (d) potential temperature anomaly (K), and (f) moisture anomaly (kg kg-1) within the radius of 5000 m cloud from the WRF model. The values in (e) and (f) are multiplied by 1 X 103 (Chen and Sun, 2002).

(b) applying a spatial filter to remove the unwanted shortwaves and the values beyond the constraints, and finally (c) introducing a polynomial or sine function as the correction function that conserves mass, while inducing the least modification to the results obtained from (a) and (b). Numerical simulations of pure advection, rotation, and idealized cyclogenesis show that the scheme is very accurate compared with analytic solutions, as shown in Fig. 19 (Sun and Sun, 2004). With the variational techniques, this scheme is capable of reproducing the positive-definite results and conservation of total mass and total energy in the shallow water equations in both rotational and irrotational frames (Sun, 2007).

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